Proof by mathematical induction

To do a proof by mathematical induction, follow the following steps exactly as shown and in theorder given:Step # 1:Show it is true for n = 1, n = 2, .......Step # 2:Suppose it is true for n = kStep # 3:Prove it is true for n = k + 1Important notes and explanations about a proof by mathematical induction:In Step # 1, you are trying to show it is true for specific values. You are free to do this test withjust one value or fifty values of your choice or more.However, showing it is true for one million values or more still does not prove it will be true for allvalues. This is a very important observation!In Step # 2, since you have already shown that it is true for one or more values, it is logical tosuppose or assume it is true for n =k or generally speaking.We usually use the asumption that we make here to complete or prove Step # 3In Step # 3, you finally show it is true for any values. Notice that step #2 did not show it is true forany values.Example:Show that for all n, 2 + 4 + 6 + ... + 2n = n ( n + 1)Step # 1:Show the equation is true for n = 1, n = 2, .......There is a pitfall to avoid here.n = 1 means the first value of the expression on the left side. In this case 2n = 2 means the first two values of the expression on the left side. In this case 2 + 4n = 3 means the first three values of the expression on the left side. In this case 2 + 4 + 6

Thus, showing the equation 2 + 4 + 6 + ... + 2n = n ( n + 1) is true for n = 4 means that we haveto show that 2 + 4 + 6 + 8 = 4 (4 + 1)2 + 4 + 6 + 8 = 6 + 6 + 8 = 12 + 8 = 20 and 4 (4 + 1) = 4 5 = 20Since the left side is equal to the right side (20 = 20) , step # 1 is done. It is not necessary tochoose other values although you could do it just for fun and to prove to yourself that it will workfor other values.Step # 2:Suppose the equation is true for n = kJust replace n by k.2 + 4 + 6 + ... + 2k = k ( k + 1)Step # 3:Prove the equation is true for n = k + 1This is the toughest part of proof by mathematical induction. Things can get really tricky here. Notin this problem though!At this point, you need to write down what it means for the equation to be true for n = k + 1Be careful! Just because you wrote down what it means does not mean that you have proved it.This is another pitfall to avoid when working on a proof by mathematical induction.Here is what it means:After you replace k by k+1, you get :2 + 4 + 6 + ... + 2 (k + 1) = k+1 ( k + 1 + 1)2 + 4 + 6 + ... + 2 ( k + 1) = k+1 ( k + 2)2 + 4 + 6 + ... + 2 ( k + 1) = ( k + 1 ) ( k + 2)Let's give you a recap because you may have lost tract of what we are trying to do here.We have not proved anything yet. The equation 2 + 4 + 6 + ... + 2 ( k + 1) = ( k + 1 ) ( k + 2)is just what it means for the equation to be true for n = k + 1We are now ready to complete the proof by mathematical induction by using the hypothesis instep # 2.starting with the hypothesis, 2 + 4 + 6 + ... + 2k = k ( k + 1)